Introduction to the monster Lie algebra. Groups, Combinatorics, and geometry, p. 99–107, edited by M. W. Liebeck and J. Saxl, L.M.S. lecture note series 165, C.U.P. 1992. Richard E. Borcherds, I would like to thank J. M. E. Hyland for suggesting many improvements to this paper. The monster sporadic simple group, of order 246320597611213317.19.23.29.31.41.47.59.71, acts on a graded vector space V = n ZVn constructed by Frenkel, Lepowsky and Meur- ⊕ ∈ man [Fre] The dimension of Vn is equal to the coefficient c(n) of the elliptic modular function j(τ) 744 = c(n)qn = q 1 + 196884q + 21493760q2 + X − − n ··· (where we write q for e2πiτ , and Im(τ) > 0). The main problem is to describe V as a graded representation of the monster, or in other words to calculate the trace Tr(g Vn) of | each element g of the monster on each space Vn. The best way to describe this information is to define the Thompson series n Tg(q) = Tr(g Vn)q X | n Z ∈ for each element g of the monster, so that our problem is to work out what these Thompson series are. For example, if 1 is the identity element of the monster then Tr(1 Vn) = | dim(Vn) = c(n), so that the Thompson series T1(q) = j(τ) 744 is the elliptic modular function. McKay, Thompson, Conway and Norton conjectured− [Con] that the Thompson series Tg(q) are always Hauptmoduls for certain modular groups of genus 0 (I will explain what this means in a moment.) In this paper we describe the proof of this in [Bor] using an infinite dimensional Lie algebra acted on by the monster called the monster Lie algebra. These Hauptmoduls are known explicitly, so this gives a complete description of V as a representation of the monster. We now recall the definition of a Hauptmodul. The group SL2(Z) acts on the upper half plane H = τ C Im(τ) > 0 by a b (τ) = aτ+b . The elliptic modular function j(τ) { ∈ | } c d cτ+d is more or less the simplest function defined on H that is invariant under SL2(Z) in much the same way that the function e2πiτ is the simplest function invariant under τ τ + 1. 1 1 7→ The element 0 1 of SL2(Z) takes τ to τ + 1, so in particular j(τ) is periodic and can be written as a Laurent series in q = e2πiτ . The exact expression for j is n 3 (1 + 240 σ3(n)q ) j(τ) = Pn>0 q (1 qn)24 Qn>0 − 3 where σ3(n) = d n d is the sum of the cubes of the divisors of n; see any book on P | modular forms or elliptic functions, for example [Ser]. Another way of thinking about j is 1 that it is an isomorphism from the quotient space H/SL2(Z) to the complex plane, which can be thought of as the Riemann sphere minus the point at infinity. There is nothing particularly special about the group SL2(Z) acting on H. We can consider functions invariant under any group G acting on H, for example some group G of finite index in SL2(Z). If G satisfies some mild conditions then the quotient H/G will again be a compact Riemann surface with a finite number of points removed. If this compact Riemann surface is a sphere, rather than something of higher genus, then we say that G is a genus 0 group. When this happens there is a more or less unique map from H/G to the complex plane C, which induces a function from H to C which is invariant under the group G acting on H. When this function is correctly normalized it is called a Hauptmodul for the genus 0 group G. For example, j(τ) 744 is the Hauptmodul for the − genus 0 group SL2(Z). a b For example, we could take G to be the group Γ0(2), where Γ0(N) = {c d ∈ SL2(Z) c 0 mod N . The quotient H/G is then a sphere with 2 points removed, so | ≡ } 1 2 that G is a genus 0 group. Its Hauptmodul starts off T2 (q) = q− + 276q 2048q + , and is equal to the Thompson series for a certain element− of the monster− of order 2··· (of type 2B in atlas notation). Similarly Γ0(N) is a genus 0 subgroup for several other values of N which correspond to elements of the monster. (However the genus of Γ0(N) tends to infinity as N increases, so there are only a finite number of integers N for which it has genus 0; more generally Thompson has shown that there are only a finite number of conjugacy classes of genus 0 subgroups of SL2(R) which are commensurable with SL2(Z).) The first person to notice any connection between the monster and genus 0 subgroups of SL2(R) was probably Ogg, who observed that the primes p for which the normalizer of Γ0(p) in SL2(R) has genus 0 are exactly those which divide the order of the monster. So the problem we want to solve is to calculate the Thompson series Tg(τ) and show that they are Hauptmoduls of genus 0 subgroups of SL2(R). The difficulty with doing this is as follows. Frenkel, Lepowsky, and Meurman constructed V as the sum of two + subspaces V and V −, which are the +1 and 1 eigenspaces of an element of order 2 in the monster. If an element g of the monster commutes− with this element of order 2, then n it is not difficult to to work out its Thompson series Tg(q) = n Tr(g Vn)q as the sum of + P | two series given by its traces on V and V − and it would be tedious but straightforward to check that these all gave Hauptmoduls. However if an element of the monster is not conjugate to something that commutes with this element of order 2 then there is no obvious + direct way of working out its Thompson series, because it muddles up V and V − in a very complicated way. We now give a bird’s eye view of the rest of this paper. We first construct a Lie algebra from V , called the monster Lie algebra. This is a generalized Kac-Moody algebra, so we recall some facts about such algebras, and in particular show how each such algebra gives an identity called its denominator formula. A well known example of this is that the denominator formulas of affine Kac-Moody algebras are the Macdonald identities. The denominator formula for the monster Lie algebra turns out to be the product formula for the elliptic modular function j, and we use this to work out the structure of the monster Lie algebra, and in particular to find its “simple roots”. Once we know the simple roots, we can write down a sort of twisted denominator formula for each element of the monster 2 group. These twisted denominator formulas imply some relations between the coefficients of the Thompson series of the monster, which are strong enough to characterize them and verify that they are indeed Hauptmoduls. We now introduce the the monster Lie algebra. This is a Z2-graded Lie algebra, 2 whose piece of degree (m, n) Z is isomorphic as a module over the monster to Vmn if (m, n) = (0, 0) and to R2 if (m,∈ n) = (0, 0), so for small degrees it looks something like 6 . 0 0 0 0 V3 V6 V9 ··· ··· 0 0 0 0 V2 V4 V6 ··· ··· 0 0 V 1 0 V1 V2 V3 ··· 0 0 0− R2 0 0 0 ··· . ··· ··· V3 V2 V1 0 V 1 0 0 ··· − ··· V6 V4 V2 0 0 0 0 ··· ··· V9 V6 V3 0 0 0 0 ··· . ··· . This is a very big Lie algebra because the dimension of Vn increases exponentially fast; most infinite dimensional Lie algebras that occur in mathematics can be graded so that the dimensions of the graded pieces have only polynomial growth. Very briefly, the construction of this Lie algebra goes as follows (this construction is not used later, so it does not matter if the reader finds this meaningless). The graded vector space V is a vertex algebra; roughly speaking this means that V has an infinite number of products on it which satisfy some rather complicated identities. (The Griess product on the 196884 dimensional space V1 is a special case of one of the vertex algebra products.) We can also construct a vertex algebra from any even lattice, and the tensor product of two vertex algebras is a vertex algebra. We take the tensor product of the monster vertex algebra V with that of the two dimensional even Lorentzian lattice (defined by the matrix 0,1 1,0 ) to get a new vertex algebra W . The monster Lie algebra is then the space of physical states of the vertex algebra W ; this is a subquotient of W , which is roughly a space of highest weight vectors for an action of the Virasoro algebra on W . The subquotient can be identified exactly using the no-ghost theorem from string theory, and turns out to be as described above. We need to know what the structure of the monster Lie algebra is. It turns out to be something called a generalized Kac-Moody algebra, so we will now have a digression to explain what these are.